Yes, it should by modern language standards, but no, that's not how the official name of the country is written.
You'll see many older names lacking the definite suffix, and this is a result of the strong Danish influence on the Norwegian language at the time. Since they're proper nouns, they're seldom changed to conform with current language norms.
In most settings, the country will be referred to as "Mexico", plain and simple.
Yes, it's confusing, but yes, there's a reason for it.
The "x" in "Mexico" is preserved because it's a proper noun, and we try not to mess too much with proper nouns - just like you wouldn't change the the spelling of someone's name based on your own pronunciation. "Meksikansk" is just an adjective, and we have less qualms with changing those to conform with Norwegian orthography.
You'll see the same thing with "Canada" and "kanadisk". Curiously, both "kanadisk" and "meksikansk" still have optional spelling variations preserving the "c" and "x" respectively - but these forms are much less used than the ones I've listed.
Maybe I'm totally confused, but is it possible to translate it as "They united Mexican states"? As in: There were several people who thought it would be nice to take some Mexican states and make one out of them. It might be completely ❤❤❤❤❤❤❤❤ in meaning (I have no clue about the history of Mexico :D), but just grammatical?